Download ( ) 2 x( )=1

Math 122
Fall 2008
Recitation Handout 2(b): Trigonometric Integrals
Evaluate each of the integrals given in the table below. In each case, describe the “trick” that is used to
convert the integral into something that you can handle more easily.
Indefinite integral
(a)
Work and equation for antiderivative
2
" x sin ( x )dx
!
Trick
Use double angle formula:
sin 2 ( x ) = 12 (1" cos(2x ))
to produce two integrals (one easy; one possible via
integration by parts)
!
"
2
(b)
3
6
# sin ( x ) cos ( x )dx
0
!
Trick
Use identity:
sin 2 ( x ) + cos2 ( x ) = 1
to create a single factor of sine or cosine (that will allow
integration by reversing the Chain Rule)
!
(c)
6
" tan( x ) sec ( x )dx
!
Re-write the integral as tan(x)sec(x) times sec5(x) to facilitate
u-substitution.
Trick
(d)
" x tan( x )dx
2
!
Trick
Use the integration formula:
" tan( x )dx = ln sec( x ) + C
(
!
)
(e)
ex
" cos e dx
( )
x
!
Trick
Use the integration formula:
" sec( x )dx = ln sec( x ) + tan( x ) + C
(
!
"
(f)
2
# cos ( x )dx
"
2
!
Trick
Use double angle formula:
cos2 ( x ) = 12 (1+ cos(2x ))
!
)
(g)
" sin(4 x ) cos(3x )dx
!
Trick
Use angle addition formulas:
sin( a + b) = sin( a) cos(b) + cos( a) sin(b)
sin( a " b) = sin( a) cos(b) " cos( a) sin(b)
to express the product as a sum of sin(7x) and sin(x).
!
!
(h)
4
" cos ( x ) sin(2x )dx
!
Trick
Use the double angle formula:
sin(2x ) = 2sin( x ) cos( x )
to turn put this entirely in terms of sin(x) and cos(x) which can
be integrated by substitution.
!
(i)
3
5
" tan ( x ) sec ( x )dx
!
Trick
Use identity:
tan 2 ( x ) + 1 = sec 2 ( x )
to express tan2(x) in terms of sec2(x), leaving a factor of
sec(x)tan(x) to accompany powers of sec(x)
!
Answers
!
!
x 2 " 18 cos(2x ) " 14 x sin(2x ) + C
(a)
1
4
(b)
2
63
(c)
1
6
sec 6 ( x ) + C
(d)
1
2
ln sec( x 2 ) + C
(e)
ln sec(e x ) + tan(e x ) + C
(f)
"
4
(g)
"1
14
cos( 7x ) " 12 cos( x ) + C
(h)
"1
3
cos 6 ( x ) + C
(i)
1
7
(
)
!
(
)
!
!
!
!
!
!
sec 7 ( x ) " 15 sec 5 ( x ) + C